Conservation of Angular Momentum 8.01 W10D2 Young and Freedman: 10.5-10.6

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Conservation of Angular Momentum8.01W10D2You
ng and Freedman 10.5-10.6
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Announcements
Math Review Night Next Tuesday from 9-11 pm Pset
10 Due Nov 15 at 9 pm No Class Friday Nov
11 Exam 3 Tuesday Nov 22 730-930 pm W011D1
Rolling without Slipping, Translation and
Rotation Reading Assignment Young and Freedman
10.3-4
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Time Derivative of Angular Momentum for a Point
Particle

• Angular momentum of a point particle about S
• Time derivative of the angular momentum about S
• Product rule
• Key Fact
• Result

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Torque and the Time Derivative of Angular
Momentum Point Particle

• Torque about a point S is equal to the time
  derivative of the angular momentum about S .

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Conservation of Angular Momentum

• Rotational dynamics
• No torques
• Change in Angular momentum is zero
• Angular Momentum is conserved

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Concept Question Change in Angular Momentum

• A person spins a tennis ball on a string in
  a horizontal circle with velocity (so that
  the axis of rotation is vertical). At the point
  indicated below, the ball is given a sharp blow
  (force ) in the forward direction. This
  causes a change in angular momentum in the
  
• direction
• direction
• direction

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Concept Question Change in Angular Momentum

• Answer 3. The torque about the center of the
  circle points in the positive -direction.
  The change in the angular momentum about the
  center of the circle is proportional to the
  angular impulse about the center of the circle
  which is in the direction of the torque

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Table Problem Cross-section for Meteor Shower

• A meteor of mass m is approaching earth as
  shown on the sketch. The distance h on the sketch
  below is called the impact parameter. The radius
  of the earth is Re. The mass of the earth is me.
  Suppose the meteor has an initial speed of v0.
  Assume that the meteor started very far away from
  the earth. Suppose the meteor just grazes the
  earth. You may ignore all other gravitational
  forces except the earth. Find the impact
  parameter h and the cross section ph2 .

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Angular Momentum for System of Particles

• Treat each particle separately
• Angular momentum for system about S

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Conservation of Angular Momentum for System of
Particles

• Assumption all internal torques cancel in pairs
• Rotational dynamics
• No external torques
• Change in Angular momentum is zero
• Angular Momentum is conserved

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Angular Impulse and Change in Angular Momentum

• Angular impulse
• Change in angular momentum
• Rotational dynamics

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Kinetic Energy of Cylindrically Symmetric Body

• A cylindrically symmetric rigid body with moment
  of inertia Iz rotating about its
• symmetry axis at a constant angular velocity
  
• Angular Momentum
• Kinetic energy

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Concept Question Figure Skater

• A figure skater stands on one spot on the ice
  (assumed frictionless) and spins around with her
  arms extended. When she pulls in her arms, she
  reduces her rotational moment of inertia and her
  angular speed increases. Assume that her angular
  momentum is constant. Compared to her initial
  rotational kinetic energy, her rotational kinetic
  energy after she has pulled in her arms must be
• the same.
• larger.
• smaller.
• not enough information is given to decide.

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Concept Question Figure Skater

• Answer 2. Call the rotation axis the z-axis.
  When she pulls her arms in there are no external
  torques in the z-direction so her z-component of
  her angular momentum about the axis of rotation
  is constant. The magnitude of her angular
  momentum about the rotation axis passing through
  the center of the stool is L I?. Her kinetic
  energy is
• Since L is constant and her moment of inertia
  decreases when she pulls her arms, her kinetic
  energy must increase.

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Constants of the Motion

• When are the quantities, angular momentum
  about a point S, energy, and momentum constant
  for a system?
• No external torques about point S angular
  momentum about S is constant
• No external work mechanical energy constant
• No external forces momentum constant

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Rotational and Translational Comparison
Quantity Rotation Translation
Force
Torque
Kinetic Energy
Momentum
Angular Momentum
Kinetic Energy
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Demo Rotating on a Chair

• A person holding dumbbells in his/her arms spins
  in a rotating stool. When he/she pulls the
  dumbbells inward, the moment of inertia changes
  and he/she spins faster.

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Concept Question Twirling Person

• A woman, holding dumbbells in her arms, spins
  on a rotating stool. When she pulls the dumbbells
  inward, her moment of inertia about the vertical
  axis passing through her center of mass changes
  and she spins faster. The magnitude of the
  angular momentum about that axis is
• the same.
• larger.
• smaller.
• not enough information is given to decide.

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Concept Question Twirling Person

• Answer 1. Call the rotation axis the z-axis.
  Because we assumed that there are no external
  torques in the z-direction acting on the woman
  then the z-component of her angular momentum
  about the axis of rotation is constant. (Note if
  we approximate the figure skater as a symmetric
  body about the z-axis than there are no
  non-z-components of the angular momentum about a
  point lying along the rotation axis.)

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Demo Train B134

• (1) At first the train is started without the
  track moving. The train and the track both move,
  one opposite the other.
• (2) Then the track is held fixed and the train
  is started. When the track is let go it does not
  revolve until the train is stopped. Then the
  track moves in the direction the train was
  moving.
• (3) Next try holding the train in place until
  the track comes up to normal speed (Its being
  driven by the train). When you let go the train
  remains in its stationary position while the
  track revolves. You shut the power off to the
  train and the train goes backwards. Put the power
  on and the train remains stationary.

A small gauge HO train is placed on a circular
track that is free to rotate.
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Table Problem Experiment 6
A steel washer, is mounted on the shaft of a
small motor. The moment of inertia of the motor
and washer is Ir. Assume that the frictional
torque on the axle is independent of angular
speed. The washer is set into motion. When it
reaches an initial angular velocity ?0, at t 0,
the power to the motor is shut off, and the
washer slows down during the time interval ?t1
ta until it reaches an angular velocity of ?a at
time ta. At that instant, a second steel washer
with a moment of inertia Iw is dropped on top of
the first washer. The collision takes place over
a time ?tcol tb - ta.

1. What is the angular deceleration a1 during the
   interval ?t1 ta?
2. What is the angular impulse due to the frictional
   torque on the axle during the collision?

c) What is the angular velocity of the two
washers immediately after the collision is
finished?
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Experiment 6 Conservation of Angular Momentum
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AppendixAngular Momentum and Torque for a
System of Particles
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Angular Momentum and Torque for a System of
Particles

• Change in total angular momentum about a point S
  equals the total torque about the point S

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Internal and External Torques

• Total external torque
• Total internal torque
• Total torque about S

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Internal Torques

• Internal forces cancel in pairs
• Does the same statement hold about pairs of
  internal torques?
• By the Third Law this sum becomes

The vector points from the
jth element to the ith element.
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Central Forces Internal Torques Cancel in Pairs

• Assume all internal forces between a pair of
  particles are directed along the line joining the
  two particles then
• With this assumption, the total torque is just
  due to the external forces
• No isolated system has been encountered such
  that the angular momentum is not constant.